Skip to main content
SpringerLink
Log in

Hybrid three-scale model for evolving pore-scale geometries

  • Original paper
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

We consider flow and upscaling of flow properties from pore scale to Darcy scale, when the pore-scale geometry is changing. The idea is to avoid having to solve for the pore evolution at the pore scale, because this results in unmanageable complexity. We propose to use stochastic modeling to parametrize plausible modifications of the pore geometry and to construct distributions of permeability parametrized by Darcy-scale variables. To localize the effects of, e.g., clogging, we introduce an intermediate scale of pore-network models. We use local Stokes solvers to calibrate the throat permeability.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Angot, P.: Analysis of singular perturbations on the Brinkman problem for fictitious domain models of viscous flows. Math. Methods Appl. Sci. 22(16), 1395–1412 (1999)

    Article  Google Scholar 

  2. Angot, P., Bruneau, C.H., Fabrie, P.: A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math. 81(4), 497–520 (1999)

    Article  Google Scholar 

  3. Arbogast, T., Brunson, D.S.: A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium. Comput. Geosci. 11(3), 207 (2007)

    Article  Google Scholar 

  4. Arbogast, T., Lehr, H.L.: Homogenization of a Darcy–Stokes system modeling vuggy porous media. Comput. Geosci. 10(3), 291–302 (2006)

    Article  Google Scholar 

  5. Auriault, J.L.: On the domain of validity of Brinkman’s equation. Transp. Porous Media 79(2), 215–223 (2009)

    Article  Google Scholar 

  6. Balhoff, M.T., Thomas, S.G., Wheeler, M.F.: Mortar coupling and upscaling of pore-scale models. Comput. Geosci. 12(1), 15–27 (2008)

    Article  Google Scholar 

  7. Bear, J.: Dynamics of Fluids in Porous Media. Dover, New York (1972)

    Google Scholar 

  8. Bear, J., Cheng, A.D.: Modeling Groundwater Flow and Contaminant Transport, vol. 23. Springer Science & Business Media (2010)

  9. Blunt, M.J.: Flow in porous media–pore-network models and multiphase flow. Curr. Opin. Colloid Interface Sci. 6(3), 197–207 (2001)

    Article  Google Scholar 

  10. Boffi, D., Brezzi, F., Fortin, M., et al.: Mixed Finite Element Methods and Applications, vol. 44. Springer, Berlin (2013)

    Book  Google Scholar 

  11. Bringedal, C., Berre, I., Pop, I.S., Radu, F.A.: A model for non-isothermal flow and mineral precipitation and dissolution in a thin strip. J. Comput. Appl. Math. 289, 346–355 (2015)

    Article  Google Scholar 

  12. Cai, R., Lindquist, W.B., Um, W., Jones, K.W.: Tomographic analysis of reactive flow induced pore structure changes in column experiments. Adv. Water Resour. 32(9), 1396–1403 (2009)

    Article  Google Scholar 

  13. Canuto, C., Kozubek, T.: A fictitious domain approach to the numerical solution of PDEs in stochastic domains. Numer. Math. 107(2), 257–293 (2007)

    Article  Google Scholar 

  14. Cesmelioglu, A., Girault, V., Riviere, B.: Time-dependent coupling of Navier–Stokes and Darcy flows. ESAIM: Math. Model. Numer. Anal. 47(2), 539–554 (2013)

    Article  Google Scholar 

  15. Chu, J., Engquist, B., Prodanovic, M., Tsai, R.: A multiscale method coupling network and continuum models in porous media i: steady-state single phase flow. Multiscale Model. Simul. 10(2), 515–549 (2012)

    Article  Google Scholar 

  16. Costa, T.B.: Hybrid multiscale methods with applications to semiconductors, porous media, and materials science. Ph.D. thesis (2016)

  17. Costa, T.B.: HybGe-Flow3D v2.0.0. https://github.com/numsol/HybGe-Flow3D (2017)

  18. Crandell, L., Peters, C.A., Um, W., Jones, K.W., Lindquist, W.B.: Changes in the pore network structure of hanford sediment after reaction with caustic tank wastes. J. Contam. Hydrol. 131(1), 89–99 (2012)

    Article  Google Scholar 

  19. Dong, H., Blunt, M.J.: Pore-network extraction from micro-computerized-tomography images. Physical Rev. E 80(3), 036307 (2009)

    Article  Google Scholar 

  20. Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements, vol. 159. Springer Science & Business Media, New York (2013)

    Google Scholar 

  21. Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. Handb. Numer. Anal. 7, 713–1018 (2000)

    Google Scholar 

  22. Gibson, N.L., Medina, F.P., Peszynska, M., Showalter, R.E.: Evolution of phase transitions in methane hydrate. J. Math. Anal. Appl. 409(2), 816–833 (2014)

    Article  Google Scholar 

  23. Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier-Stokes Equations(book) (Lecture Notes in Mathematics), vol. 749. Springer, Berlin (1979)

    Book  Google Scholar 

  24. Gordon, A., Powell, C.E.: A preconditioner for fictitious domain formulations of elliptic PDEs on uncertain parameterized domains. SIAM/ASA J. Uncertain. Quantif. 2(1), 622–646 (2014)

    Article  Google Scholar 

  25. Harbrecht, H., Peters, M.: Combination technique based second moment analysis for PDEs on random domains. In: Sparse Grids and Applications - Stuttgart, pp 51–77. Springer, Cham (2014)

  26. Harbrecht, H., Peters, M., Siebenmorgen, M.: Numerical solution of elliptic diffusion problems on random domains. Preprint 2014-08, Mathematisches Institut, Universität Basel (2014)

  27. Hassanizadeh, M., Gray, W.G.: General conservation equations for multi-phase systems: 1. Averaging procedure. Adv. Water Resour. 2, 131–144 (1979)

    Article  Google Scholar 

  28. Hassanizadeh, M., Gray, W.G.: General conservation equations for multi-phase systems: 2. Mass, momenta, energy, and entropy equations. Adv. Water Resour. 2, 191–203 (1979)

    Article  Google Scholar 

  29. Hornung, U.: Homogenization and Porous Media, vol. 6. Springer Science & Business Media, New Year (2012)

    Google Scholar 

  30. Kim, D., Peters, C., Lindquist, W.: Upscaling geochemical reaction rates accompanying acidic CO2-saturated brine flow in sandstone aquifers. Water Resour. Res. 47(1) (2011)

  31. Krotkiewski, M., Ligaarden, I.S., Lie, K.A., Schmid, D.W.: On the importance of the Stokes-Brinkman equations for computing effective permeability in karst reservoirs. Commun. Comput. Phys. 10(05), 1315–1332 (2011)

    Article  Google Scholar 

  32. Labs, P.: Paralution v1.0.0. http://www.paralution.com/ (2015)

  33. Lindquist, W.B., Lee, S., Oh, W., Venkatarangan, A., Shin, H., Prodanovic, M.: 3DMA-Rock: a software package for automated analysis of rock pore structure in 3-D computed microtomography images. SUNY Stony Brook (2005)

  34. Lukarski, M.S.D.: Parallel sparse linear algebra for multi-core and many-core platforms. Ph.D. thesis, Georgia Institute of Technology (2012)

  35. Mahabadi, N., Dai, S., Seol, Y., Sup Yun, T., Jang, J.: The water retention curve and relative permeability for gas production from hydrate-bearing sediments: pore-network model simulation. Geochem. Geophys. Geosyst. 17(8), 3099–3110 (2016)

    Article  Google Scholar 

  36. Medina, F.P., Peszynska, M.: Hybrid modeling and analysis of multicomponent adsorption with applications to coalbed methane. In: Wolfe, D. (ed.) Porous Media: Theory, Properties, and Applications, chapter 1, pp 1–52. Nova, Commack (2016)

  37. Mikelic, A., Jäger, W.: On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60(4), 1111–1127 (2000)

    Article  Google Scholar 

  38. Mittal, R., Iaccarino, G.: Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239–261 (2005)

    Article  Google Scholar 

  39. Ossiander, M., Peszynska, M., Vasylkivska, V.: Conditional Stochastic Simulations of Flow and Transport with Karhunen-Loeve Expansions, Stochastic Collocation, and Sequential Gaussian Simulation. J. Appl. Math. 2014 (652594), 21 (2014). https://doi.org/10.1155/2014/652594

    Google Scholar 

  40. Ossiander, M., Peszynska, M., Madsen, L., Mur, A., Harbert, W.: Estimation and simulation for geospatial porosity and permeability data. Environ. Ecol. Stat. 24, 109 (2017). https://doi.org/10.1007/s10651-016-0362-y

    Article  Google Scholar 

  41. Patankar, S.: Numerical heat transfer and fluid flow. CRC Press, Boca Raton (1980)

    Google Scholar 

  42. Peskin, C.S.: The immersed boundary method. Acta Numer. 11, 479–517 (2002)

    Article  Google Scholar 

  43. Peszyńska, M., Wheeler, M.F., Yotov, I.: Mortar upscaling for multiphase flow in porous media. Comput. Geosci. 6(1), 73–100 (2002)

    Article  Google Scholar 

  44. Peszyńska, M., Trykozko, A., Augustson, K.: Computational upscaling of inertia effects from porescale to mesoscale, pp 695–704. Springer, Berlin (2009)

    Google Scholar 

  45. Peszynska, M., Trykozko, A.: Convergence and Stability in Upscaling of Flow with Inertia from Porescale to Mesoscale. Int. J. Multiscale Comput. Eng. 9(2), 215–229 (2011). https://doi.org/10.1615/IntJMultCompEng.v9.i2.60

    Article  Google Scholar 

  46. Peszynska, M., Trykozko, A.: Pore-to-core simulations of flow with large velocities using continuum models and imaging data. Comput. Geosci. 17(4), 623–645 (2013)

    Article  Google Scholar 

  47. Peszynska, M., Trykozko, A., Iltis, G., Schlueter, S., Wildenschild, D.: Biofilm growth in porous media: experiments, computational modeling at the porescale, and upscaling. Adv. Water Resour. 95, 288–301 (2016)

    Article  Google Scholar 

  48. Prodanović, M., Lindquist, W., Seright, R.: 3D image-based characterization of fluid displacement in a Berea core. Adv. Water Resour. 30(2), 214–226 (2007)

    Article  Google Scholar 

  49. Pruess, K., Garcia, J.: Multiphase flow dynamics during CO2 disposal into saline aquifers. Environ. Geol. 42(2), 282–295 (2002)

    Article  Google Scholar 

  50. Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. CMCS-BOOK-2009-019. Oxford University Press, Oxford (1999)

    Google Scholar 

  51. Quarteroni, A., Manzoni, A., Negri, F.: Reduced Basis Methods for Partial Differential Equations: an Introduction, vol. 92. Springer, Berlin (2015)

    Google Scholar 

  52. Raoof, A., Hassanizadeh, S.M., Leijnse, A.: Upscaling transport of adsorbing solutes in porous media: pore-network modeling. Vadose Zone J. 9(3), 624–636 (2010)

    Article  Google Scholar 

  53. Raoof, A., Nick, H., Hassanizadeh, S., Spiers, C.: Poreflow: a complex pore-network model for simulation of reactive transport in variably saturated porous media. Comput. Geosci. 61, 160–174 (2013)

    Article  Google Scholar 

  54. Russell, T.F., Wheeler, M.F.: Finite element and finite difference methods for continuous flows in porous media. In: The Mathematics of Reservoir Simulation, vol. 1, pp 35–106 (1983)

  55. Sánchez-Palencia, E.: Non-homogeneous media and vibration theory. In: Non-homogeneous Media and Vibration Theory, vol. 127 (1980)

  56. Schulz, R., Ray, N., Frank, F., Mahato, H., Knabner, P.: Strong solvability up to clogging of an effective diffusion–precipitation model in an evolving porous medium. Eur. J. Appl. Math. 28(2), 179–207 (2017)

    Article  Google Scholar 

  57. Silin, D., Patzek, T.: Pore space morphology analysis using maximal inscribed spheres. Phys. A: Stat. Mech. Appl. 371(2), 336–360 (2006)

    Article  Google Scholar 

  58. Spiteri, E., Juanes, R., Blunt, M.J., Orr, F.M., et al.: Relative-permeability hysteresis: trapping models and application to geological CO2 sequestration. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2005)

  59. Tartar, L.: Convergence of the homogenization process. In: Nonhomogeneous Media and Vibration Theory (1980)

  60. Trykozko, A., Brouwer, G., Zijl, W.: Downscaling: a complement to homogenization. Int. J. Numer. Anal. Model. 5, 157–170 (2008)

    Google Scholar 

  61. Trykozko, A., Zijl, W., Bossavit, A.: Nodal and mixed finite elements for the numerical homogenization of 3D permeability. Comput. Geosci. 5(1), 61–84 (2001)

    Article  Google Scholar 

  62. van Noorden, T.L., Pop, I., Ebigbo, A., Helmig, R.: An upscaled model for biofilm growth in a thin strip. Water Resour. Res. 46(6) (2010)

  63. Versteeg, H.K., Malalasekera, W.: An Introduction to Computational Fluid Dynamics: the Finite Volume Method. Pearson Education (2007)

  64. Vo, H.X., Durlofsky, L.J.: A new differentiable parameterization based on principal component analysis for the low-dimensional representation of complex geological models. Math. Geosci. 46(7), 775–813 (2014)

    Article  Google Scholar 

  65. Weinan, E., Engquist, B., Li, X., Ren, W., Vanden-Eijnden, E.: Heterogeneous multiscale methods: a review. Commun. Comput. Phys. 2(3), 367–450 (2007)

    Google Scholar 

  66. Whitaker, S.: Volume averaging of transport equations. In: Prieur du Plessis, J (ed.) Fluid Transport in Porous Media. Computational Mechanics Publications, Southampton (1997)

  67. Xiu, D.: Numerical Methods for Stochastic Computations: a Spectral Method Approach. Princeton University Press, Princeton (2010)

    Google Scholar 

  68. Xiu, D., Tartakovsky, D.M.: Numerical methods for differential equations in random domains. SIAM J. Sci. Comput. 28(3), 1167–1185 (2006)

    Article  Google Scholar 

  69. Zijl, W., Trykozko, A.: Numerical homogenization of the absolute permeability using the conformal-nodal and mixed-hybrid finite element method. Transp. Porous Media 44(1), 33–62 (2001)

    Article  Google Scholar 

Download references

Acknowledgements

Wewould like to gratefully acknowledge the collaborations with two colleagues, and our financial support. We also appreciate the comments from anonymous reviewers which helped to improve this manuscript.

Dr Anna Trykozko from University of Warsaw helped us to validate the software HybGe-Flow3D by running comparisons of permeabilities we obtained with those reported in [46], which were based on flow simulations with ANSYS Fluent.

We also want to thank Dr Masa Prodanovic from The University of Texas at Austin who helped us with the software 3DMA-Rock for pore–network extraction [33].

Funding

This work was partially supported by the grants NSF DMS-1115827 and NSF DMS-1522734. This research was partially supported by the grants NSF DMS-1115827 and NSF DMS-1522734.

Author information

Authors and Affiliations

  1. Numerical Solutions, Inc, Corvallis, OR, USA

    Timothy B. Costa

  2. Department of Mathematics, Oregon State University, Corvallis, OR, USA

    Timothy B. Costa, Kenneth Kennedy & Malgorzata Peszynska

Authors
  1. Timothy B. Costa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kenneth Kennedy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Malgorzata Peszynska
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Timothy B. Costa.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Costa, T.B., Kennedy, K. & Peszynska, M. Hybrid three-scale model for evolving pore-scale geometries. Comput Geosci 22, 925–950 (2018). https://doi.org/10.1007/s10596-018-9733-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-018-9733-9

Keywords

Search

Navigation